Frege’s conceptual notation includes a significant revaluation of the
grammatical role of the copula and, by extension, of the natural language
intransitive verb, examples of which are “exists” or “there are.” Upon
careful inspection the predicate position in any sentence is shown to be a function sign whose argument
place could be
filled by either a name or a compound expression for a concept and a
variable. The value of the predicate is a concept and not a particular.
"Exists" stands for an operator and not a predicate and so it is a mistake to
say that it is a symbol of a concept. In Frege’s own example (Begriffsschrift
p. 23), “There is a house” becomes “Something is a house (There is at least
one house),” which can be understood as a function mapping (or subsuming)
whatever satisfies its argument onto (or under) the concept, “…is a house.”
Because this function is the truth functional equivalent to, “It is not the
case that everything is not a house,” its expression can be replaced
wherever it occurs by the latter without changing the meanings of the
propositional functions involved.

This sharpened
understanding of the “exists” or “is” verb is radically consequential for
philosophical issues involving the existence of individual objects or
classes of objects, and it is equally consequential for theological “proofs”
that rely on the assumption that that verb is completely meaningful by
itself, bereft of any supplementary meaning provided by a predicate concept.

Accordingly
one wonders what Frege himself may have thought of non-logically motivated
definitions of existence such as “To exist is to be perceived,” or proofs
like the so-called ontological proof of the existence of god, or even
propositions to the effect that god exists.

One text from
the Nachlass
entitled Dialog mit Pünjer Über Existenz contains a number of
appropriate and enlightening remarks. Frege addresses the issue of whether
existence can be defined as perceptibility or indeed as limited by any other
predicate that means something more than the existential operator. In this
dialogue Frege makes three points:

A)When we
say that, as in the case of hallucinations or imaginary constructs, no
object caused our mental idea (Vorstellung) (e.g. our idea of a
centaur), the subject of our proposition is “mental idea of a centaur” and
not “centaur.” This is a second-order proposition about ideas or
representations; when we say something caused our idea of Peter, for
example, our proposition predicates “corresponding to something” of ideas.

B)When we
say, “Men exist” we are saying that it is not the case that (using standard
notation) for any a whatsoever, “a is not a man” is true.

C)When we
utter a statement including a referring term as its subject, we assume the
object referred to exists, and our assumption is “self-evident” (selbstverständlich).
At the same time, the words “is,” “exists,” “there is,” etc. have no meaning
(keinen Inhalt). They are merely grammatical placeholders like
punctuation marks. One way of understanding this is that they tell us no
more about the subject of the sentence than that it is identical to itself.
Any attempt to define existence terms of this sort with words that do have
meaningful predicative content (such as “experientiable”) leads to
contradiction.

(C)
constitutes his objection to the doctrine that proposes that to exist means
to be perceived or possibly perceived. For, (C) entails that certain
statements about unperceived (or imperceptible) objects such as imaginary
objects are self-contradictory. For example, by substitution of expressions
with identical meanings, the following propositions should all mean the same
thing, that is they should all have the same truth values.

1)Centaurs
are imaginary.

2)Centaurs
cannot be perceived.

3)There are
centaurs that cannot be perceived.

4)There are
centaurs that do not exist.

5)Some
centaurs exist that do not exist.

So by (C)
perceptibility, or indeed any predicate that similarly limits the meaning of
existence, cannot be a valid definition of “exists.” Frege’s solution to
this confusion is (A).

Frege’s
conclusion has consequences for those philosophers, dating roughly from
Locke onwards, for whom
perceptibility is an important qualification on what can be known, and, in
certain respects, also on what exists. One of the goals of
modernphilosophy has been to put a cap on what were viewed as wild conclusions
about strange abstractions, conclusions drawn with virtually no
justification beyond conformity to a few rules of deduction. And, because
the abstractions involved were so poorly defined or ill conceived, they gave
rise to the suspicion that they in fact meant nothing at all. The challenge
was to give content to this suspicion.
Bacon and
Descartes first laid down the
principle that real knowledge begins with what we sense; real knowledge is
directly or indirectly knowledge about what we perceive or can perceive.
Locke elaborated the Cartesian ground rule into a richer theory of ideas
that presented a mechanism whereby sensations and other ideas constitute and
provide the material for knowledge. Good Xtians that they were, Descartes
and Locke excepted beliefs based on revelation as well as certain sorts of
proof, mainly those having to do with god, from the Cartesian ground rule.
Kant made at least two
contributions to this critical function of modern philosophy. First, he
lifted the exception of proofs about god from the Cartesian ground rule and
subjected proofs of god’s existence, even those advocated by Descartes (an
ontological proof that
throws in causality for good measure) and Locke (the
cosmological proof), to
detailed criticism. Secondly Kant interpreted the Cartesian rule about what
could be known as a set of putative
rules about what
exists. In Pünjer’s formulation, the principle that knowledge begins
with sensation turns into a basic postulate that only things that directly
or indirectly cause or could cause sensations do in fact exist. (Pünjer, of
course, is not representative, of what Kant really said, for Kant began to
distinguish between types of objects such that, in what comes down to a
tautology, the perceptibility criterion was limited to objects of sense.)

Does Frege’s
attack on Pünjer really target and thereby undo the critical conclusions of
modern philosophers? Not really. Descartes and Locke limited their criticism
to what could be known and did not in any clear way relate it to ontological
issues. And Frege finds a way to formulate Kant’s insight that “existence is
not a real predicate,” without embracing Kant’s view about faculties and
ideas or indeed any theory of mind at all. He does this by creating a
symbolic notation where the function of the copula and most of the natural
language meanings of “exists’ are performed by the existential operator, an
expression that can be read as stating, “This sentence is about….” (That is
what Frege meant when he said that the imputation of existence in a
well-formed proposition is selbstverständlich.)

Frege may not
have had an explicit grasp of what the empiricists were trying to achieve
when they linked knowledge (and sometimes existence) to what we perceive.
(For example, Frege cites numerous philosophers’ views on the definition of
number in the Grundlagen,
but all his references come from one book, Baumann’s.) Empirically oriented
philosophers want to prevent meaningless conclusions drawn by logical rules
concerning meaningless concepts invented by the metaphysicians and
schoolmen. Nevertheless, by not keeping an eye on the empiricist goal, Frege
could produce a dispassionate analysis of what we really mean when we say
some (mental) ideas are caused by external objects and some are not.

On p. 20 of
the Nachlass Frege makes comments about what we mean when we argue
about the existence of specific things that are not literally wrong but they
do betray a lack of understanding of empiricist philosophies. He observes
that the scientists who claimed that there exists a planet beyond Uranus did
not mean that their idea of this planet was caused by an object acting on
their senses (auf Grund einer Affektion des Ich), viz. that they
actually perceived this planet. It is true that Leverrier did not say a
ninth planet exists because he actually saw it (although Galileo said that
Jupiter has moons because he actually saw them). However, Leverrier did rely
on direct observation of the deviations in the orbit of Uranus that he felt
were best explained by the existence of a ninth planet and its gravitational
influence. And Leverrier meant (unless he was completely off the rails)
that, were we to have telescopes strong enough, we would actually see the
ninth planet, which indeed turned out to be the case. Leverrier’s hypothesis
was what some would call an empirical claim. Namely it involved actual
observations and perceptions of some sort and the prediction that, when
certain conditions are met, people would actually perceive the ninth planet.
What Leverrier probably meant when he said a ninth planet existed was not
that it was perceived or perceivable, but that some body possessing mass was
located in space and time such as to affect the orbit of Uranus. What
empirically minded philosophers wanted to do was to distinguish what they
considered legitimate issues such as the existence of Uranus from fanciful
issues such as the Platonic chestnut that the Good as such exists. Their
approach was to argue that legitimate questions about existence are related
to what we perceive or what we can perceive. Despite the troubles with
empiricist theories of mind, the need for making some such distinction
remains relevant if we are to avoid uncontrolled speculation.

On the same
page Frege uses exactly the same phrase concerning the existence of god.
What those who affirm or deny the existence of god mean, he says, is not
that their idea of god is or can be caused by the perception of an external
object. Frege takes care to add that he means a direct perception (unmittelbare
Affektion) of god. Indeed the Xtian definition of god is that thing
whose essence is to exist. It is notably lacking in instructions as to where
to find it. Likewise, the notion of existence as a predicate is critical to
the proofs of the existence of god as proposed by
Anselm,
Descartes, Spinoza and
Leibniz. So, while Xtian philosophers do not mean that god’s existence is
tantamount to god’s perceivability, they do mean that god’s existence can be
proved and that what is thereby proved says something more about god than
the form of a proposition says about any argument that can occupy the
subject position of that proposition.

The result is
that one of the wild abstractions referred to above is god as defined by
Xtian philosophers and theologians. And indeed Kant ties empiricist views
and the ontological proof together nicely in the celebrated passage from his
Critique of Pure Reason.

Frege endorses
Kant’s observation that existence is not a real predicate with the attendant
consequences that the propositions that make up the ontological proof of the
existence of god are grammatically faulty and so meaningless. In fact Frege
provides a basis for this observation that is much stronger than Kant’s
thaler example. For, if existence is a real predicate, then a proposition
like

6)God
exists.

Should be
grammatically indistinguishable from a simple predicate proposition like

7)
God is
stupid.

But they are
not. (7) can be analyzed as,

8)There is
a god who is stupid.

But (6)
becomes,

9)There is
a god that is.

Which sounds a
bit like burning bush talk. Just as Frege coyly asserts that any proposition
that denies existence to imperceptible objects contradicts itself, so (9) is
redundant. Even this empty sense of existence as applying to anything you
care to name or subsume under a predicate can be analyzed in such a way that
existential propositions appear to be no more than a particular form of
general proposition where the natural language words, “exist,” “is,” “there
is” etc. don’t appear at all. In other words:

10) x(Fx)
≡ ¬(x)­­(¬Fx).

That is: An F
exists if and only if it is not the case that everything is not an F. Under
this analysis, to say that god exists is to say that it is not the case that
everything is not god. That may or may not be true, just as it may or may
not be true that it is not the case that everything is not a centaur. But
the ontological proof purports to prove that (6) is not only true, it is
necessarily true, whatever that means. The ontological proof incorporates
propositions that require that existence be put back in the predicate
position: “god’s essence is to exist,” “god has all the perfections
including existence,” “Whatever other properties god may have, existence is
one of them.” These quaint formulations all share a requirement for a
stronger predicative sense of existence than the weak form that just serves
to introduce any proposition whatsoever. Under Fregean analysis, any
proposition that includes a distinct predicate “exists” that nevertheless
means the same thing as the logical operator, “There is…,” dissolves into
absurdity. But any proposition that includes a distinct predicate “exists”
that means something more than or different from the logical operator both
leads to contradictions as shown above and risks defining “god” in such a
way as would be unacceptable to the proponents of the ontological proof.

The first
thing is to see what happens when we try to insert the contextually defined
version of “exists” into propositions in places where genuine first-order
predicates (“is green,” “is stupid” etc. would normally be inserted). Begin
with a simple assertion of existence:

11) There are centaurs.

Now let us
replace “x is a centaur” with “Fx.” (11) now reads

12)
x(Fx).

By (10),
therefore,

13) It is not
the case that everything is not a centaur.

The working
assumption is that “…exists” is a predicate just like “…is a centaur.” That
is “x exists” can be substituted for “x is a centaur” in any well-formed
proposition and the result remains a well-formed proposition subject to the
same logical rules of substitution as propositions involving, “x is a
centaur.” Let “Gx” stand for “x exists.” Now the proposition that x exists
can be written,

14)
x(Gx).

By (10), (14)
can be rewritten

15) ¬(x)(¬Gx).

By Frege’s
analysis of “exists”,

16) “an F
exists” =def “x(Fx)”

So, by (10)

17) “a G exists”
=def ¬(x)(¬Gx).

Accordingly ¬(x)(¬Gx)
can be substituted in (15) where Gx occurs. The result is

18) ¬(x)¬ (¬ (x)(¬Gx)).

But “Gx”
appears in this proposition also and so we should be able to substitute (15)
for “Gx” and so on ad infinitum. The result is not a proposition, but an
infinite iteration of the incomplete propositional fragment, “It is not the
case that, for anything whatsoever, it is not the case that….” The upshot is
that by, using Fregean notation, which is in this respect no more than a
clarification of our concepts, the superficially meaningful proposition
(14), “Something exists,” is not a proposition at all.

All the
ontological proofs involve as steps propositions that are identical to (14)
or are like (14) or presuppose (14) or entail (14). Take the proposition
that is a step in both Descartes’ and Anselm’s versions of the ontological
proof: god’s essence is to exist (This statement can be found on p. 1137 of
Descartes’ Letters). Restated without the concept of essence, this
proposition states that, whatever other properties may be attributed to god,
existence is one of them. Existing is incontrovertibly one of god’s
properties. Anything that has all of god’s other properties but does not
exist is not god. Let F1…F2 be a set of divine
properties (It need not be complete, but it can be). Let G be the property
“exists” as in our previous paragraph. So if a=god, then,

19) F1a
&…&Fna &Ga.

Substituting
for Ga in this proposition according to (10) and (17), we get,

If we
interpret “God’s essence is existence” to mean that “to be god” entails “to
exist,” we get the same result.

23) (x)((x is
god) Gx)

becomes

24) (x)((x is
god) ¬(x)(¬(Gx))

and so on.

A foolish
person might try to extricate himself from this unsatisfactory iteration by
choosing to regard the logical operator “exists” as different from the
predicate “exists” with the conclusion that the two can both be used in a
proposition. The hoped for result would be well formed propositions that
state that god exists. Frege shows us why such a person would be foolish
when he points out that the negations of propositions asserting that there
are objects that cannot be perceived are self-contradictory. For, if
“perceptibility” means the same thing as “exists,” then such propositions
come down to saying, “Something exists that does not exist.” In the same way
every true statement containing only the existential operator and the
foolish person’s existential predicate, viz. any proposition of the form,
x(x exists), would be a tautology
true of everything, and the negation of any such proposition would be a
contradiction true of nothing.

That same
foolish person, or perhaps someone like-minded, might try to assert that one
can get by with just the existential operator. That is he might say that
“There is a god” (and that god has all these F’s) is a true statement, or
perhaps a “necessarily” true statement, and he might try to pimp the
ontological proof to demonstrate that it is necessarily true. The problem is
that Frege’s existential operator is no more than a grammatical placeholder
introducing a proposition. It has no meaning, as Frege says. The negation of
x(Fx), for example, does not state
that x does not exist. It states that x is not F. The existential operator
is just a live and let live kind of guy. As far as it is concerned, god
exists, Zeus exists, Yahweh exists, Aphrodite exists and so does Moombalah
the Terminator. And for my money, it’s far preferable for Aphrodite than for
Yahweh to exist. After all, she’s just out of the bath, she’s got perky
nipples and she’s ready for action. Much better than some smelly old bearded
character who’s been hiding out in the desert.

Frege’s
conceptual notation does not by itself refute the ontological proof of the
existence of god. Nor is it meant to do so. The importance of the conceptual
notation is that it allows us to express clearly and with perspicuous
deductive consequences propositions in which we refer to things (It is
important for Frege’s conceptual notation that every well formed sentence -
except sentences where “exists,” as well as perhaps certain other
“transcendental” terms from traditional metaphysics, is forced into the
predicate position – can be fully and accurately transcribed) and, most
importantly, propositions in which, using natural language, we assert that
things do or do not exist. All of those situations in which we are concerned
about what we had gathered together in the catch-all verb “exists” –
situations involving our doubts or hopes about centaurs, or murder weapons
or tenth planets – are much more perspicuously stated by replacing the verb
“exists” by real predicates, such as “romping on the slopes of Olympus,” or
“lying in the butler’s pantry,” or “describing orbit O.” Frege’s actual
refutations come in his comments not only in his dialogue with the
unfortunate Pünjer, but also in the Grundlagen. Indeed
just stating what our real meaning when we use a word might be, is not in
itself a complete argument. Someone might simply say they don’t agree that
that is the real use, or else that they choose to use the word with some
specialized sense. So when we clarify the use of a term from natural
language or philosophical terminology, we need to bring the argument home,
so to speak, by showing how the “unreal” use leads to contradictions, for
example, or an inability to answer a simple question like, “What do you
mean” (Uncle Ben’s inability
to say what he means when he says, “Doo-li exists, all right,” is an example
of this.)

On pp. 86-87
of the Grundlagen Frege gives a different formulation of the
fallacies of the ontological proof. He begins by noting the similarity (Ähnlichkeit)
between the concept of existence and he concept of number. Affirmation that
and F exists is a denial that the number of things falling under the concept
F is zero. Existence is a qualification of a concept and characteristics of
concepts do not qualify the objects that fall under those concepts (although
conclusions may be drawn about the objects from the fact that the concepts
are characterized in a certain way). The fallacy behind the ontological
proof is that it tries to apply a concept-specific quality to a particular,
namely god. For the same reason the theological assertion that god is one is
also a grammatical fallacy. The only thing troubling about this formulation
is that to assert that existence, like number, is a qualification of
concepts is not the same as to assert that existence in properly formulated
propositions about particulars is either meaningless (keinen Inhalt)
or self-evident (selbstverständlich). The most Frege can say is that
the meaning of an existential proposition is contained in (10).

The other
issue that the meaning of existence could have a bearing on is the one of
the ontological status of various objects: mental objects, ideal objects
etc. In one passage (p. 16) Frege seems to deny credence that existence is
some sort of overarching concept (übergeordnete Begriff), a highest
Aristotelian category. He equates existence with self-identity and states
that it has no meaning. It functions as part of the form of a sentence.
Derivative from this notion is the quasi-concept of entity (Seiendes) .
No conclusions can be drawn about what this quasi-concept or its equivalents
(e.g. Sein) could stand for. “Wenn die Philosophen von dem
‘absoluten Sein’ sprechen, so ist dies eigentlich eine Vergötterung der
Kopula.” (p. 17) Certainly no one wants to deify the copula. That
stirring issue deserves its own discussion.

Note.
In other passages Frege asserts that existence is a kind of highest
concept that subsumes (übergeordnet) other predicative concepts like
being human (pp. 20-21). This position runs into Russell’s problems with
universal sets which would be reflected in the failure of Basic Law V from
Frege’s Grundgesetze and Heidegger’s frustration with coming up with
a definition of being in general using “logical” thought. Instead of dealing
with that complex topic in detail here, it is worthwhile to make two modest
points. First, it is not clear that defining existence as a highest
all-subsuming concept or category is compatible with saying that it is
meaningless. For it could be argued that the meaningful content (Inhalt)
of a concept is at least partially constituted by the meaningful content of
the concepts it subsumes. For example, if the concept mammal subsumes the
concept dog, then it could be argued that at least part of the meaning of
“mammal” is “possibly being a dog.” Secondly the distinct doctrines that (a)
the meaning of “exists” and related terms is determined by the form of the
proposition and not because they are predicated of the proposition’s
argument, and (b) existence is a concept that subsumes all other predicative
concepts – these distinct doctrines do not entail each other. At least (a)
does not entail (b). For example, the meaning of the propositional
conjunction sign is part of the form of propositions containing as
determined by truth functions. But conjunction is not a highest concept
subsuming other concepts. So it is possible to retain Frege’s genuine
insight into the meaning of existence terms and the consequences of that
insight as described above without thereby being committed to treating
existence as some sort of super category.